Local Geometry of Self-similar Sets: Typical Balls, Tangent Measures and Asymptotic Spectra.
We analyze the local geometric structure of self-similar sets with open set condition through the study of the properties of a distinguished family of spherical neighborhoods, the typical balls. We quantify the complexity of the local geometry of self-similar sets, showing that there are uncountably...
| Authors: | , , |
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| Format: | article |
| Publication Date: | 2023 |
| Country: | España |
| Institution: | Universidad Complutense de Madrid (UCM) |
| Repository: | Docta Complutense |
| Language: | English |
| OAI Identifier: | oai:docta.ucm.es:20.500.14352/94994 |
| Online Access: | https://hdl.handle.net/20.500.14352/94994 |
| Access Level: | Open access |
| Keyword: | 5 Self-similar Sets Hausdorff Measures Tangent Measures Density of Measures Computability of Fractal Measures Complexity of Topological Spaces Sierpiński Gasket Matemáticas (Matemáticas) Geometría Análisis matemático 12 Matemáticas 1204 Geometría 1202 Análisis y Análisis Funcional |
| Summary: | We analyze the local geometric structure of self-similar sets with open set condition through the study of the properties of a distinguished family of spherical neighborhoods, the typical balls. We quantify the complexity of the local geometry of self-similar sets, showing that there are uncountably many classes of spherical neighborhoods that are not equivalent under similitudes. We show that at a tangent level, the uniformity of the Euclidean space is recuperated in the sense that any typical ball is a tangent measure of the measure at mu-a.e. point, where mu is any self-similar measure. We characterize the spectrum of asymptotic densities of metric measures in terms of the packing and centered Hausdorff measures. As an example, we compute the spectrum of asymptotic densities of the Sierpiński gasket. |
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